Temporal planning (TP) is notoriously difficult because it requires to solve a propositional STRIPS planning problem with temporal constraints. In this paper, we propose an efficient strategy for solving TP, which combines, in an innovative way, several well established and studied techniques in AI, OR and constraint programming. Our approach integrates graph planning (a well studied planning paradigm), max-SAT (a constraint optimization technique), and the Program Evaluation and Review Technique (PERT), a well established technique in OR. Our method first separates the logical and temporal constraints of a TP problem and solves it in two phases. In the first phase, we apply our new STRIPS planner to generate a parallel STRIPS plan with a minimum number of parallel steps. Our new STRIPS planner is based on a new max-SAT formulation, which leads to an effective incremental learning scheme and a goal-oriented variable selection heuristic. The new STRIPS planner can generate optimal parallel plans more efficiently than the well-known SATPLAN approach. In the second phase, we apply PERT to schedule the activities in a parallel plan to create a shortest temporal plan given the STRIPS plan. When applied to the first optimal parallel STRIPS plan, this simple strategy produces optimal temporal plans on most benchmarks we have tested. This strategy can also be applied to optimal STRIPS plans of different parallel steps in an anytime fashion to find an optimal temporal plan. Our experimental results show that the new strategy is effective and the resulting algorithm outperforms many existing temporal planners.


Constraint Programming Hybrid Strategy Temporal Plan Bound Model Check Temporal Planner 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
  2. 2.
  3. 3.
    Benedetti, M., Bernardini, S.: Incremental compilation-to-SAT procedures. In: H. Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 46–58. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Blum, A., Furst, M.L.: Fast planning through planning graph analysis. Artificial Intelligence 90, 281–300 (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bonet, B., Geffner, H.: Planning as heuristic search. Artificial Intelligence, Special issue on Heuristic Search 129(1) (2001)Google Scholar
  6. 6.
  7. 7.
  8. 8.
    Edelkamp, S.: Mixed propositional and numerical planning in the model checking integrated planning system. In: Proceedings of AIPS 2002, Workshop on Planning for Temporal Domains, pp. 47–55 (2002)Google Scholar
  9. 9.
    Fikes, R.E., Nilsson, N.J.: STRIPS: A new approach to the application of theorem proving to problem solving. Artificial Intelligence 2, 189–208 (1971)CrossRefzbMATHGoogle Scholar
  10. 10.
    Awedh, M., Jin, H., Somenzi, F.: CirCus: a satisfiability solver geared towards bounded model checking. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 519–522. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Haslum, P.: TP4’04 and HSP. In: Proceedings of IPC4, ICAPS, pp. 38–40 (2004)Google Scholar
  12. 12.
    Hillier, F., Lieberman, G.: Introduction to Operations Research, 7th edn. McGraw-Hill, Boston (2001)zbMATHGoogle Scholar
  13. 13.
    Hoffmann, J., Nebel, B.: The FF planning system: Fast plan generation through heuristic search. Journal of Artificial Intelligence Research 14, 253–302 (2001)zbMATHGoogle Scholar
  14. 14.
    Hooker, J.N., Vinay, V.: Branching rules for satisfiability. Journal of Automated Reasoning 15, 359–383 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Katukam, S., Kambhampati, S.: Learning explanation-based search control rules for partial order planning. In: Proceedings of AAAI 1994, pp. 582–587 (1994)Google Scholar
  16. 16.
    Kautz, H.: SATPLAN04: Planning as satisfiability. In: Proceedings of IPC4, ICAPS (2004)Google Scholar
  17. 17.
    Kautz, H., Selman, B.: Unifying SAT-based and graph-based planning. In: Proceedings of IJCAI 1999, pp. 318–325 (1999)Google Scholar
  18. 18.
    Koehler, J., Hoffmann, J.: On reasonable and forced goal orderings and their use in an agenda-driven planning algorithm. Journal of Artificial Intelligence Research 12, 338–386 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    McDermott, D.: Estimated-regression planning for interactions with web services. In: Proceedings of AIPS 2002, pp. 204–211 (2002)Google Scholar
  20. 20.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of the 38th Design Automation Conference (DAC 2001) (2001)Google Scholar
  21. 21.
    Nigenda, R.S., Nguyen, X., Kambhampati, S.: AltAlt: Combining the advantages of Graphplan and heuristic state search. Technical report, Arizona State University (2000)Google Scholar
  22. 22.
    Pollock, J.L.: The logical foundations of goal-regression planning in autonomous agents. Artificial Intelligence 106(2), 267–334 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Refanidis, I., Vlahavas, I.: The GRT planner. AI Magazine, 63–66 (2001)Google Scholar
  24. 24.
    Ryan, L.: Efficient algorithms for clause-learning SAT solvers. Master’s thesis, Simon Fraser University (2003)Google Scholar
  25. 25.
    Selman, B., Kautz, H.: Planning as satisfiability. In: Proceedings ECAI 1992, pp. 359–363 (1992)Google Scholar
  26. 26.
    Shtrichman, O.: Tuning SAT checkers for bounded model checking. In: Computer Aided Verification, pp. 480–494 (2000)Google Scholar
  27. 27.
    Vidal, V., Geffner, H.: CPT: An optimal temporal POCL planner based on constraint programming. In: Proceedings of IPC4, ICAPS, pp. 59–60 (2004)Google Scholar
  28. 28.
    Wolfman, S., Weld, D.: Combining linear programming and satisfiability solving for resource planning. The Knowledge Engineering Review 15(1) (2000)Google Scholar
  29. 29.
    Hsu, C., Chen, Y., Wah, B.W.: SGPlan: Subgoal partitioning and resolution in planning. In: Proceedings of IPC4, ICAPS, pp. 30–32 (2004)Google Scholar
  30. 30.
    Zhang, L., Madigan, C.F., Moskewicz, M.W., Malik, S.: Efficient conflict driven learning in boolean satisfiability solver. In: ICCAD, pp. 279–285 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zhao Xing
    • 1
  • Yixin Chen
    • 1
  • Weixiong Zhang
    • 1
  1. 1.Department of Computer Science and EngineeringWashington University in Saint LouisSaint LouisUSA

Personalised recommendations